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Annales scientifiques de l'ENS - Titles - série 4, 51 (2018)

Titles < série 4, 51

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 51, fascicule 4 (2018)

Alexis Drouot
Scattering resonances for highly oscillatory potentials
Annales scientifiques de l'ENS 51, fascicule 4 (2018), 865-925

Télécharger cet article : Fichier PDF

Résumé :
Résonances de potentiels rapidement oscillants
Nous étudions les résonances de potentiels à support compact V_(x) = W(x, x/), où W : R ^d R ^d/(2Z )^d C et d est impair. Ainsi, V_ est la somme d'un potentiel qui varie lentement W_0 et d'un potentiel qui oscille à fréquence 1/. Quand W_0 0 nous prouvons que V_ n'a pas de résonances dans la zone Im -A (^-1) mise à part une unique résonance proche de 0 si d = 1. Nous montrons par un exemple explicite que ce résultat est optimal. Cela prouve une conjecture de Duchêne-Vukićević-Weinstein [12]. Quand W_0 0 et W est lisse nous montrons que les resonances de V_ qui restent bornées lorsque tend vers 0 admettent une expansion en puissances de . Les arguments de la preuve permettent de calculer les coefficients de cette expansion. Nous construisons un potentiel effectif qui converge uniformément vers W_0 lorsque tend vers 0 et dont les résonances sont à distance O(^4) de celles de W_0. Cela améliore et étend les résultats de Duchêne, Vukićević et Weinstein à toutes les dimensions impaires.

Mots-clefs : Résonances, potentiels rapidement oscillants, expansions asymptotiques.

Abstract:
We study resonances of compactly supported potentials V_(x) = W ( x, x/) where W : R ^d R ^d / ( 2Z ) ^d C, d odd. That means that V_ is a sum of a slowly varying potential, W_0, and one oscillating at frequency 1/. When W_0 0 we prove that there are no resonances above the line Im= -A (^-1), except a simple resonance near 0 when d=1. We show that this result is optimal by constructing a one-dimensional example. This settles a conjecture of Duchêne-Vukićević-Weinstein [12]. When W_0 0 and W smooth we prove that resonances in fixed strips admit an expansion in powers of . The argument provides a method for computing the coefficients of the expansion. We produce an effective potential converging uniformly to W_0 as 0 and whose resonances approach resonances of V_ modulo O(^4). This improves the one-dimensional result of Duchêne, Vukićević and Weinstein and extends it to all odd dimensions.

Keywords: Scattering resonances, highly oscillatory potentials, asymptotic expansions.

Class. math. : 35P15, 35P25, 42B20.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Blanes, Sergio and Casas, Fernando
On the convergence and optimization of the Baker-Campbell-Hausdorff formula
Linear Algebra Appl. 378 (2004) 135–158
Math Reviews MR2031789
2
Borisov, D. I.
On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential
J. Math. Sci. (N.Y.) 139 (2006) 6243–6322
Math Reviews MR2278906
3
Borisov, D. I.
On some singular perturbations of periodic operators
Teoret. Mat. Fiz. 151 (2007) 207–218
Math Reviews MR2338076
4
Borisov, D. I. and Gadyl'shin, R. R.
On the spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential
Teoret. Mat. Fiz. 147 (2006) 58–63
Math Reviews MR2254715
5
Christiansen, T.
Schrödinger operators with complex-valued potentials and no resonances
Duke Math. J. 133 (2006) 313–323
Math Reviews MR2225694
6
Corless, R. M. and Gonnet, G. H. and Hare, D. E. G. and Jeffrey, D. J. and Knuth, D. E.
On the Lambert W function
Adv. Comput. Math. 5 (1996) 329–359
Math Reviews MR1414285
7
Dimassi, Mouez
Semi-classical asymptotics for the Schrödinger operator with oscillating decaying potential
Canad. Math. Bull. 59 (2016) 734–747
Math Reviews MR3563753
8
Dimassi, Mouez and Duong, Anh Tuan
Scattering and semi-classical asymptotics for periodic Schrödinger operators with oscillating decaying potential
Math. J. Okayama Univ. 59 (2017) 149–174
Math Reviews MR3643435
9
Drouot, Alexis
Bound states for highly oscillatory potentials in dimension 2
SIAM J. Math. Anal. 50 (2018) 1471–1484
10
11
12
Duchêne, Vincent and Vukićević, Iva and Weinstein, Michael I.
Scattering and localization properties of highly oscillatory potentials
Comm. Pure Appl. Math. 67 (2014) 83–128
Math Reviews MR3139427
13
Duchêne, Vincent and Vukićević, Iva and Weinstein, Michael I.
Homogenized description of defect modes in periodic structures with localized defects
Commun. Math. Sci. 13 (2015) 777–823
Math Reviews MR3318385
14
Duchêne, Vincent and Weinstein, Michael I.
Scattering, homogenization, and interface effects for oscillatory potentials with strong singularities
Multiscale Model. Simul. 9 (2011) 1017–1063
Math Reviews MR2831589
15
16
Gesztesy, F. and Latushkin, Y. and Mitrea, M. and Zinchenko, M.
Nonselfadjoint operators, infinite determinants, and some applications
Russ. J. Math. Phys. 12 (2005) 443–471
Math Reviews MR2201310
17
Golowich, S. E. and Weinstein, Michael I.
Scattering resonances of microstructures and homogenization theory
Multiscale Model. Simul. 3 (2005) 477–521
Math Reviews MR2136162
18
Klopp, Frédéric
Resonances for ``large'' ergodic systems in one dimension: a review
in Spectral analysis of quantum Hamiltonians
Oper. Theory Adv. Appl. 224 (2012) 171–182
Math Reviews MR2962860
19
Klopp, Frédéric
Resonances for large one-dimensional ``ergodic'' systems
Anal. PDE 9 (2016) 259–352
Math Reviews MR3513136
20
21
22
Sá Barreto, Antônio and Zworski, Maciej
Existence of resonances in potential scattering
Comm. Pure Appl. Math. 49 (1996) 1271–1280
Math Reviews MR1414586
23
Simon, Barry
The bound state of weakly coupled Schrödinger operators in one and two dimensions
Ann. Physics 97 (1976) 279–288
Math Reviews MR0404846
24
Simon, Barry
Notes on infinite determinants of Hilbert space operators
Advances in Math. 24 (1977) 244–273
Math Reviews MR0482328
25
Titchmarsh, E. C.
The theory of functions
Oxford Univ. Press, Oxford, 1939
Math Reviews MR3728294
26
Smith, Hart F. and Zworski, Maciej
Heat traces and existence of scattering resonances for bounded potentials
Ann. Inst. Fourier (Grenoble) 66 (2016) 455–475
Math Reviews MR3477881
27
Zworski, Maciej
Sharp polynomial bounds on the number of scattering poles
Duke Math. J. 59 (1989) 311–323
Math Reviews MR1016891